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    \section*{Comprehensive Overview}

    During my undergraduate studies, I maintained stable academic performance, particularly excelling in mathematics-related courses. Specifically, I achieved an average score of above 85 in courses such as Mathematical Analysis, Complex Functions, Discrete Mathematics, and Numerical Analysis, with the latter scoring as high as 92. These courses not only solidified my foundation in mathematics but also honed my logical thinking and problem-solving skills. Moreover, I dedicated substantial time and effort to study advanced mathematical concepts like special functions, infinite series, and non-elementary integrals, reading extensively and engaging with many experts in the field, demonstrating my profound interest in and enthusiasm for mathematics.

    In addition, I participated in a thesis project during my undergraduate studies and devised my own research topic, culminating in a formal paper. This indicates my capability to conduct independent academic research and articulate research findings clearly. I invested significant effort and time into researching and exploring for my paper, which was composed of my own independent findings, involving extensive calculations and derivations. The logical clarity and articulate presentation of my paper evidenced my abilities in independent thinking and exploration, as well as in academic writing, earning recognition from my faculty during my thesis defense.

    \section*{Specific Examples}

    My insight into infinite series is also keen. Once, when Professor Chen assigned a problem to prove the existence of a limit of a recursive relation and find it, I not only quickly solved the problem but also recognized it as a generalization of the well-known Viete’s infinite product, which I had read about in extracurricular books. Unsatisfied with stopping there, I discovered several similar forms of infinite products and series, formulating a general generating formula for them. Some examples include:
    \begin{gather*}
        \prod_{n = 1}^{\infty} \frac{1 + \sec \left( \frac{x}{2^n} \right)}{2} = \frac{2\tan \left( \frac{x}{2} \right)}{x}, \\
        \prod_{n = 1}^{\infty} \left( 2 - \sec \left( \frac{x}{2^n} \right) \right) = \frac{1}{3} x \csc(x) \left( 1 + 2 \cos(x) \right), \\
        \prod_{n = 1}^{\infty} \left( 1 - \left( 1 - \sec \left( \frac{x}{2^n} \right)^2 \right)^2 \right) = x^3 \cot(x) \csc(x)^3, \\
        \prod_{n = 1}^{\infty} \left( 1 - \left( 1 - \sec \left( \frac{x}{2^n} \right) \right)^2 \right) = \frac{1}{3} x^2 \csc(x)^2 (1 + 2 \cos(x)).
    \end{gather*}

    On another occasion, right after learning about indefinite integral calculations in class, I eagerly applied them to my series calculations. Through a day's effort, I successfully solved integrals such as $ \int \frac{2 \cos \left( \frac{1}{3} \arcsin \left( \frac{3 \sqrt{3 x^3}}{2} \right) \right)}{\sqrt{4 - 27 x^3}} dx $ and $ \int \frac{\sqrt{\sqrt{16 x^4 + 1} + 1}}{\sqrt{2} \sqrt{16 x^4 + 1}} dx $, deriving several formulas related to $\pi$, such as:
    \begin{gather*}
        \sum_{n=0}^{\infty } \binom{3n}{n}\left( \frac{64}{729} \right)^{n} \left(\frac{3}{3n+1} + \frac{4}{3n+2} \right) = \frac{3\sqrt{3}\pi}{4} + \frac{3}{2}, \\
        \sum_{n=0}^{\infty } (-1)^{n} \binom{4n}{2n}\left( \frac{9}{256} \right)^{n} \left(\frac{8}{4n+1} + \frac{3}{2n+1} - \frac{12}{4n+3} \right) = \frac{8\sqrt6\pi}{9}.
    \end{gather*}

    I also used definite integral transformations and polynomial division to arrive at the following combinatorial number series:
    \begin{gather*}
        \sum_{n=0}^{\infty } \frac{8^{n}}{\binom{4n}{n} } \left( \frac{13}{4n+1} - \frac{4}{2n+1} + \frac{5}{4n+3} \right) = 8\pi, \\
        \sum_{n=0}^{\infty } \frac{(-1)^{n}\binom{2n}{n} }{3^n\binom{3n}{n}\binom{6n}{3n}  } \left( \frac{131}{6n+1} - \frac{1}{6n+5} \right) = 24\sqrt{3}\pi, \\
        \sum_{n=0}^{\infty } \frac{(-1)^{n}}{72^{n}\binom{7n}{3n}} \left( \frac{369756}{7n+1} - \frac{12759}{7n+2} + \frac{8920}{7n+3} - \frac{2070}{7n+4} + \frac{156}{7n+5} - \frac{187}{7n+6} \right) = \frac{201684\sqrt{3}\pi}{3}.
    \end{gather*}

    My exploratory spirit was again commended by Professor Chen.

    \section*{My Strengths Include}

    \begin{enumerate}
        \item \textbf{Outstanding Academic Performance:} I have excelled in mathematics-related courses, demonstrating a deep understanding and solid mastery of the subject.
        \item \textbf{Research Interest and Capability:} I have a strong interest in advanced mathematical concepts such as special functions, infinite series, and non-elementary integrals, and I spend considerable time outside of class on research and exploration, showing my capability for independent thinking and exploration.
        \item \textbf{Independent Research and Writing Ability:} I participated in the graduation thesis project, devised my own research topic, and completed a formal paper, demonstrating my ability to conduct independent academic research and clearly express research findings.
        \item \textbf{Extensive Learning and Exploration Experience:} Through reading a vast number of papers and discussing with professors, I continually expand my academic horizons, showing good self-learning capabilities and an exploratory spirit.
        \item \textbf{Mathematical Tool Application Ability:} I am proficient in using professional software like Wolfram Alpha, Desmos, Geogebra, etc., effectively employing these tools for mathematical modeling and analysis, which enhances my efficiency and accuracy in academic and professional tasks.
    \end{enumerate}

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